Stronger distortion theorems of univalent functions and its applications (Q793177)

In this paper, by means of the representation theorem of continuous linear functionals on the space of continuous functions, the author improves and generalizes the famous FitzGerald inequalities, the Bazilevich inequalities and the Hayman regular theorem. For example: Theorem III.1. Suppose \(f\in S\), \(\{\gamma_{k,n}\}\) are the Grunsky coefficients of \(g(s)=1/f(z)=\zeta +b_ 1\zeta^{-1}+b_ 2\zeta^{- 2}+...,\quad z=1/\zeta,\quad \theta_ 0=\theta_ 0(f)\) and \(\alpha_ f\) are the Hayman direction and Hayman constant of f respectively, \(\ln(f(z)/z)=2(\gamma_ 1z+\gamma_ 2z^ 2+...).\) If \(\sum^{\infty}_{k=1}| \eta_ k|^ 2<\infty,\) then we have the sharp inequality \[ (frac{1}{2}\ln 1/\alpha_ f- \sum^{\infty}_{n=1}n| \gamma_ n-(1/n)\exp(-in \theta_ 0)|^ 2) \] \[ \times(\sum^{\infty}_{k=1}(1/k)| \eta_ k|^ 2-\sum^{\infty}_{n=1}n| \sum^{\infty}_{k=1}\eta_ k\gamma_{k,n}|^ 2) \] \[ \geq | \sum^{\infty}_{k=1}\eta_ k[((1/k)\exp(-ik \theta_ 0)-\gamma_ k)+\sum^{\infty}_{n=1}n\overline{(\gamma_ n-(1/n)\exp(-in \theta_ 0))}\gamma_ n|^ 2. \] Theorem V.1. Suppose \(F(\zeta)=\zeta +b_ 1\zeta^{-1}+b_ 2\zeta^{-2}+...\in \Sigma ',\quad G(w)=F^{- 1}(w)=w+B_ 1w^{-1}+B_ 2w^{-2}+...,\) and \(\{\lambda_{p,q}\}\) are the Grunsky coefficients. Then we have \[ | \sum^{\infty}_{m,n=1}\lambda_{m,n}\chi_ m\chi_ n| \leq \sum^{\infty}_{n=1}(1/n)| \chi_ n+\sum^{\infty}_{m=n+2}B_ m^{(-n)}\chi_ m|^ 2, \] where \(B_ m^{(-n)}\) is defined as \([G(w)]^{-k}=w^{-k}+\sum^{\infty}_{n=k+2}B_ n^{(-k)}w^{- n}.\)